Finite Group Databasebeta

← #27281792

Subgroups — #27281792

Order 13568.

554 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283744C211yesmaximal, characteristic
2#20273432C212yesmaximal, complemented
3#20273056C211yesmaximal, characteristic
4#20272571C212yesmaximal, complemented
5C53×#2542C211yesmaximal, characteristic
6#16810695—24no—
7#16810680C2212yes—
8#16810680C2212yes—
9#16810587—21no—
10C2×C22⋊#88322—21no—
11#16810369—24no—
12#16810347—21no—
13#16810347—21no—
14C2×C4⋊Q8×C53C2211yescharacteristic
15C2×C22⋊Q8×C53—21no—
16C2×C22⋊Q8×C53—21no—
17C2×C4×D4×C53C2211yescharacteristic
18C53×#399—21no—
19C53×#399—21no—
20C53×#390C412yes—
21C53×#384C412yes—
22C4×C4⋊C4×C53C2211yescharacteristic
23C2×#11719917—22no—
24C22×#88322—22no—
25C4⋊#88322—22no—
26C4⋊#88322D414yes—
27C22⋊#88322—22no—
28C22⋊#88322—22no—
29C22⋊#88322—22no—
30C22⋊#88322—22no—
31#13133128—22no—
32#13133128—22no—
33C22×Q8×C53—21no—
34C22×D4×C53C2×C411yescharacteristic
35C23×C4×C53D411yescharacteristic
36C23×C4×C53D411yescharacteristic
37C4⋊Q8×C53—22no—
38C4⋊Q8×C53—22no—
39C22⋊Q8×C53—44no—
40C22⋊Q8×C53—44no—
41C4×D4×C53—22no—
42C4×D4×C53—22no—
43C2×C4⋊C4×C53—22no—
44C2×C4⋊C4×C53—22no—
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53—21no—
47C2×C4⋊C4×C53C2×C412yes—
48C2×C4⋊C4×C53C2×C411yescharacteristic
49C2×C22⋊C4×C53—22no—
50C2×C22⋊C4×C53—22no—
51C2×C22⋊C4×C53D411yescharacteristic
52C2×C22⋊C4×C53D411yescharacteristic
53C2×C42×C53C2×C412yes—
54C2×C42×C53C2311yescharacteristic
55#95×C53—21no—
56C24×C53C22⋊C411yescharacteristic
57C24×C53C22⋊C411yescharacteristic
58C2×Q8×C53—42no—
59C2×Q8×C53—42no—
60C2×Q8×C53—41no—
61C2×Q8×C53—41no—
62C2×D4×C53—22no—
63C2×D4×C53—22no—
64C2×D4×C53—41no—
65C22×C4×C53—22no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—22no—
71C22×C4×C53—42no—
72C22×C4×C53—42no—
73C22×C4×C53C22⋊C411yescharacteristic
74C22×C4×C53C22⋊C411yescharacteristic
75C22×C4×C53C22×C411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C4⋊C4×C53—24no—
79C4⋊C4×C53—22no—
80C4⋊C4×C53—24no—
81C4⋊C4×C53—44no—
82C4⋊C4×C53—44no—
83C4⋊C4×C53—42no—
84C22⋊C4×C53—22no—
85C22⋊C4×C53—22no—
86C22⋊C4×C53—44no—
87C22⋊C4×C53—44no—
88C42×C53—42no—
89C42×C53C2×D412yes—
90C42×C53#4112yes—
91#11719917—82no—
92C2×#88322—24no—
93C2×#88322—42no—
94C2×#88322—82no—
95C23×C53—21no—
96C23×C53—21no—
97C23×C53—21no—
98C23×C53—21no—
99C23×C53—21no—
100C23×C53—41no—
101C23×C53—41no—
102C23×C53—41no—
103C23×C53—41no—
104C23×C53#10112yes—
105C23×C53C2×C22⋊C411yescharacteristic
106Q8×C53—82no—
107Q8×C53—82no—
108D4×C53—44no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—22no—
112C2×C4×C53—24no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—41no—
116C2×C4×C53—42no—
117C2×C4×C53—42no—
118C2×C4×C53—42no—
119C2×C4×C53—42no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—42no—
123C2×C4×C53—42no—
124C2×C4×C53—42no—
125C2×C4×C53—81no—
126C2×C4×C53—81no—
127C2×C4×C53C4×D412yes—
128C2×C4×C53#12011yescharacteristic
129C2×C4×C53#12112yes—
130C2×C4×C53#12311yescharacteristic
131#88322—44no—
132#88322—82no—
133#9455—531nomaximal, Sylow
134C22×C53—21no—
135C22×C53—21no—
136C22×C53—22no—
137C22×C53—42no—
138C22×C53—42no—
139C22×C53—41no—
140C22×C53—41no—
141C22×C53—41no—
142C22×C53—41no—
143C22×C53—41no—
144C22×C53—81no—
145C22×C53—81no—
146C22×C53#38611yescharacteristic
147C22×C53C2×#10111yescharacteristic
148C22×C53#41311yescharacteristic
149C4×C53—21no—
150C4×C53—22no—
151C4×C53—22no—
152C4×C53—44no—
153C4×C53—41no—
154C4×C53—41no—
155C4×C53—82no—
156C4×C53#45312yes—
157#2973—531no—
158#2542D5311yescharacteristic
159#1556—532no—
160#1466—531no—
161#1285—532no—
162C2×C53—21no—
163C2×C53—21no—
164C2×C53—42no—
165C2×C53—81no—
166C2×C53#188711yescharacteristic
167C2×C53#190111yescharacteristic
168C2×C53#190911yescharacteristic
169C2×C4⋊Q8C2×D5311yescharacteristic
170C2×C22⋊Q8—21no—
171C2×C22⋊Q8—21no—
172C2×C4×D4C2×D5311yescharacteristic
173#435—1064no—
174#432—532no—
175#432—532no—
176#407—1061no—
177C2×C22⋊C8—1061no—
178#399—21no—
179#399—21no—
180#390Dic5312yes—
181#384Dic5312yes—
182C4×C4⋊C4C2×D5311yescharacteristic
183#332—1064no—
184#323—1061no—
185#323—1061no—
186C53#945511yescharacteristic, complemented, Sylow
187C22×Q8—21no—
188C22×D4C2×Dic5311yescharacteristic
189C23×C4C22⋊D5311yescharacteristic
190C23×C4C22⋊D5311yescharacteristic
191C2×M16—1062no—
192C22×C8—1062no—
193C4⋊Q8—22no—
194C4⋊Q8—22no—
195C22⋊Q8—44no—
196C22⋊Q8—44no—
197C4×D4—22no—
198C4×D4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4C2×Dic5312yes—
204C2×C4⋊C4C2×Dic5311yescharacteristic
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4C22⋊D5311yescharacteristic
208C2×C22⋊C4C22⋊D5311yescharacteristic
209C2×C42C2×Dic5312yes—
210C2×C42C22×D5311yescharacteristic
211C4⋊C8—534no—
212C4⋊C8—1062no—
213C22⋊C8—1062no—
214C22⋊C8—1062no—
215C22⋊C8—1062no—
216C22⋊C8—1062no—
217#97—1062no—
218#97—1062no—
219#95—21no—
220C24C22⋊Dic5311yescharacteristic
221C24C22⋊Dic5311yescharacteristic
222C2×Q8—42no—
223C2×Q8—42no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—41no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4C22⋊Dic5311yescharacteristic
238C22×C4C22⋊Dic5311yescharacteristic
239C22×C4C22×Dic5311yescharacteristic
240C22×C4C2×C22⋊D5311yescharacteristic
241C22×C4C2×C22⋊D5311yescharacteristic
242M16—4242no—
243C2×C8—1064no—
244C2×C8—4242no—
245C2×C8—2122no—
246C4⋊C4—24no—
247C4⋊C4—22no—
248C4⋊C4—24no—
249C4⋊C4—44no—
250C4⋊C4—44no—
251C4⋊C4—42no—
252C22⋊C4—22no—
253C22⋊C4—22no—
254C22⋊C4—44no—
255C22⋊C4—44no—
256C42—42no—
257C42D4×D5312yes—
258C42#1171994712yes—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23#1313313612yes—
269C23C2×C22⋊Dic5311yescharacteristic
270Q8—82no—
271Q8—82no—
272D4—44no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—24no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—41no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4D4×Dic5312yes—
292C2×C4#1313318511yescharacteristic
293C2×C4#1313318912yes—
294C2×C4#1313319811yescharacteristic
295C8—4242no—
296C8—2124no—
297C22—21no—
298C22—21no—
299C22—22no—
300C22—42no—
301C22—42no—
302C22—41no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—81no—
308C22—81no—
309C22#1681050311yescharacteristic
310C22C2×#1313313611yescharacteristic
311C22#1681060711yescharacteristic
312C4—22no—
313C4—21no—
314C4—22no—
315C4—44no—
316C4—41no—
317C4—41no—
318C4—82no—
319C4#1681079012yes—
320C2—21no—
321C2—21no—
322C2—42no—
323C2—81no—
324C2#2027511411yescharacteristic
325C2#2027518411yescharacteristic
326C2#2027523811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2542C2
socleC22×C53#386
prime radicalC22×D4×C53C2×C4
cosocleC2×C42×C53C23
hypercenter#2542D53
hypocenterC53#9455
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2542

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #20273432C2semidirect
4. #20272571C2semidirect
186. C53#9455semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202837442
#202734322
#202730562
#202725712
C53×#25422
#945553